Transcription of The Multivariate Gaussian Distribution
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The Multivariate Gaussian Distribution
The Multivariate Gaussian DistributionChuong B. DoOctober 10, 2008A vector-valued random variableX= X1 Xn Tis said to have amultivariatenormal (or Gaussian ) distributionwith mean Rnand covariance matrix Sn++1if its probability density function2is given byp(x; , ) =1(2 )n/2| |1/2exp 12(x )T 1(x ) .We write this asX N( , ). In these notes, we describe Multivariate Gaussians and someof their basic Relationship to univariate GaussiansRecall that the density function of aunivariate normal (or Gaussian ) distributionisgiven byp(x; , 2) =1 2 exp 12 2(x )2 .Here, the argument of the exponential function, 12 2(x )2, is a quadratic function of thevariablex. Furthermore, the parabola points downwards, as the coefficient of the quadraticterm is negative.
Recall that the density function of a univariate normal (or Gaussian) distribution is given by p(x;µ,σ2) = 1 √ 2πσ exp − 1 2σ2 (x−µ)2 . Here, the argument of the exponential function, − 1 2σ2(x−µ) 2, is a quadratic function of the variable x. Furthermore, the parabola points downwards, as the coefficient of the quadratic term ...
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